An electronic nematic liquid in BaNi2As2

Understanding the organizing principles of interacting electrons and the emergence of novel electronic phases is a central endeavor of condensed matter physics. Electronic nematicity, in which the discrete rotational symmetry in the electron fluid is broken while the translational one remains unaffected, is a prominent example of such a phase. It has proven ubiquitous in correlated electron systems, and is of prime importance to understand Fe-based superconductors. Here, we find that fluctuations of such broken symmetry are exceptionally strong over an extended temperature range above phase transitions in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{{{{\rm{Ba}}}}}}{{{{{{\rm{Ni}}}}}}}_{2}{({{{{{{\rm{As}}}}}}}_{1-x}{{{{{{\rm{P}}}}}}}_{x})}_{2}$$\end{document}BaNi2(As1−xPx)2, the nickel homologue to the Fe-based systems. This lends support to a type of electronic nematicity, dynamical in nature, which exhibits a particularly strong coupling to the underlying crystal lattice. Fluctuations between degenerate nematic configurations cause splitting of phonon lines, without lifting degeneracies nor breaking symmetries, akin to spin liquids in magnetic systems.


Dilatometry and transport
Electrical resistance was measured in standard 4-wires geometry, either using a combination of LakeShore 6221 current source and LakeShore 2182A Nanovoltmeter, or using a high precision resistance bridge (LakeShore 372). Typical samples dimensions were 2mm x 1mm x 50 µm. Resistance was measured along the a crystallographic direction of the tetragonal unit cell. A slow cooling rate of about 1K to 1.5K was used to cool down the single crystals. To avoid any unwanted external stress on the sample, measurements were performed under fully free-standing conditions. In Supplementary Figure 2-a, we compare the resistivity of the BaNi 2 (As 1−x P x ) 2 sample with x=0.035 to high-resolution thermal expansion ∆L/L obtained on a similar sample (upon warming). Details for the dilatometry measurements are given in ref. 1 . The dilatometry measured along the [100] and the [110] direction (in the high-T tetragonal notation) shows a pronounced first-order transition below T Tr i = 105 K. Below T ρ = 127 K, a clear difference between the ∆L/L measured along the two directions is seen and disappear gradually. This evidences a second order transition from a weakly distorted orthorhombic phase that exists above T Tr i and up to T ρ , above which the tetragonal phase is restored.
In Supplementary Figure 2-b, we show that the splitting of this orthorhombic onset coincides with the broad minimum in the temperature derivative of the resistivity dR/dT. Note that this coincidence seems to hold primarily at low doping. Preliminary dilatometry investigation in the 7.6% sample indicate that the orthorhombic transition occurs at a temperature lower than T ρ and could not be detected in the 10% sample, even though the system displays an I-CDW and a minimum in dR/dT consistent with the other samples (hence the different symbol for T ρ for this sample in Fig. 5). This will be investigated further independently but does not affect any of the conclusions of the present work. A complete thermodynamic study of BaNi 2 (As 1−x P x ) 2 is presented in ref. 2 .   temperature T c for BaNi 2 (As 1−x P x ) 2 with x = 0, 0.073, and 0.1.

Heat Capacity
To measure the superconducting T c , we have mounted the BaNi 2 (As 1−x P x ) 2 samples (x =

Supplementary Note 3. Raman scattering
Raman scattering measurements were performed on single crystals of BaNi 2 (As,P) 2 with the low-resolution mode of a single grating Jobin-Yvon Labram spectrometer (600 grooves/mm grating) to maximize the signal output. Phonon measurements were done with a He-Ne laser (633 nm). The Raman-active optical modes of BaNi 2 (As,P) 2 were determined following the nuclear site group analysis. Each unit cell contains one Ba-, two As-and two Ni-atoms, resulting in 15 zone-center vibrational modes. In Supplementary Table I Triclinic phase (P-1) Atom Site symmetry Irreducible representations Supplementary Tab. I: Atomic site symmetries and the resulting normal modes for each point and the Raman-active optical modes Γ Raman for BaNi 2 (As,P) 2 in different phases.

Group theory and polarization selection rules
At room temperature, BaNi 2 (As,P) 2 possesses the tetragonal structure with space group I4/mmm and point group D 4h , thus 1A 1g , 1B 1g and 2E g modes are expected.
Upon cooling, the system undergoes a second-order orthorhombic transition and the space group changes to I/mmm with point group D 2h 1 . By breaking the four-fold symmetry, the A 1g and B 1g modes of the tetragonal phase become A g modes in the orthorhombic phase, and the doubly degenerate E g modes of the tetragonal phase split into nondegenerate B 2g and B 3g modes in the orthorhombic phase. However, the amplitude of the orthorhombic distortion is  Table II. Theoretical estimations of the phonon energies are given in Supplementary Note 3.2. Note that due to the small amplitude of the orthorhombic distortion and the narrow temperature window of the orthorhombic phase, the phonon frequencies are expected to be close to those in the tetragonal phase.

Phonon energy calculation
Lattice dynamics properties for the for the different structures of BaNi 2 As 2 structure were calculated using the linear response or density-functional perturbation theory (DFPT) implemented in the mixed-basis pseudopotential method.
The electron-ion interaction is described by norm-conserving pseudopotentials, which were constructed following the descriptions of Hamann, Schlüter, Chiang 3,4 for Ba and Vanderbilt 5 for Ni and As, respectively. Semi-core states Ba-5p, Ni-3s, Ni-3p were included in the valence space. In the mixed-basis approach, valence states are expanded in a combination of plane waves and local functions at atomic sites, which allows an efficient description of more localized components of the valence states. Here, plane waves with a cut-off for the kinetic energy For both E g modes, the symmetry reduction induced by the orthorhombic distortion leads to splittings of less than 0.3 cm −1 .

Fitting details
The Raman spectra in the XZ geometry were fitted between 10 cm −1 and 1220 cm −1 . To subtract the continuum a background with the analytical formula only for x = 0.076 and x = 0.10 a second damped harmonic oscillator profile was added.
To check the robustness of the fit different approaches for the background were tested all giving comparable peak positions/splitting and relative intensities.

E g mode broadening, splitting and determination of T *
The most spectacular result of our study is the large splitting of the E g phonons upon cooling.
To determine the onset temperature of the effect, T * in the phase diagram shown in Fig. 5 of the main text, we have employed two methods which yield very similar results. On the one hand, T * corresponds to the onset of the broadening to the E g,1 mode upon cooling, which can be directly extracted from the fitting of the data. In Supplementary Figure 6-a, we show the (background subtracted) low energy data taken in BaNi 2 As 2 with the XZ polarization, which already evidences the anomalous softening of the mode with decreasing temperature. The E g,1 phonons have been aligned by shifting the spectra relative to the energy of the mode and their intensity normalized, from which a broadening is clearly visible already at 180 K ( Fig. S6-b).
The linewidth (full width at half maximimum) of the E g,1 is ploted as function of temperature Supplementary Fig. 7: a) Temperature dependence of the splitting amplitude (black points) and broadening (blue points) of the E g,1 mode of BaNi 2 As 2 . The former are obtained from the two mode fitting (see e.g. Supplementary Figure 5-b), whereas the latter data is obtained form a single line fit (Supplementary Figure 5-a). b) same for BaNi 2 (As 1−x P x ) 2 (x=0.035) c) same for BaNi 2 (As 1−x P x ) 2 (x=0.076) d) same for BaNi 2 (As 1−x P x ) 2 (x=0.1). In the panels b, c and d, the grey shaded area corresponds to the temperature region where the two lines cannot be fully resolved.
in Supplementary Figure 7 for the four investigated samples. At temperatures low enough, the splitting is unambiguous (the grey shaded areas of panels b,c and d of that figure correspond to regions where the phonon lineshape deviates from standard single damped harmonic oscillator and can already be fitted with two modes, even though the two peaks cannot be resolved) and increases linearly as temperature decreases. When extrapolating this linear dependence to high temperatures, we find that the temperature at which the splitting vanishes corresponds well to T * determined from the broadening.
Supplementary Fig. 8: a) Temperature dependence of the ZZ Raman spectra of BaNi 2 As 2 . All spectra are scaled and shifted vertically for clarity. b) XX and ZZ Raman spectra of BaNi 2 As 2 at 5K. The dashed lines represent the calculated phonon frequencies for the triclinic phase (see Supplementary Note 3.2).

Spectra in the triclinic phase
The symmetry reduction of across the first order structural transition is accompanied with a strong renormalization of the vibrational spectra. This is well illustrated by the spectra obtained in the ZZ polarization configuration, which probes the A g symmetry.
As discussed in Supplementary Note 3.1, only 6 Raman-active modes with the A g symmetry are expected below the first-order triclinic transition. However, as can be seen in Supplementary   Figure 8, much more modes can be resolved in the triclinic phase and are related to the formation of the commensurate CDW superstructure. Note that we also observe a significant renormalization of the electronic background across the triclinic transition over a broad spectral range, which will be discussed in a future publication.

Raman measurements under stress
The behavior of the E g modes in strained pure BaNi 2 As 2 is discussed in the main text. A crystal of BaNi 2 As 2 was broken in half, and we glued one of half on a glass-fiber-reinforced plastic (GFRP) substrate that was used to detwin BaFe 2 As 2 in thermal expansion measurements 8 .
The glue was chosen to be the Devcon 5 Minute Epoxy to ensure strong adhesion. The other half was glued on a copper piece with high-vacuum Apiezon N Grease as strain-free reference.
The mounting configuration is shown in Supplementary Figure 9. The sample was required to be very flat and thin for a homogeneous strain field along the crystallographic c-axis. The width and height of the fiberglass were specifically cut to 1 mm, and the whole piece of fiberglass was glued on the cold finger of our cryostat with Apiezon to ensure good thermal contact. The solid lines in Supplementary Figure 9(a) indicate the fiber direction. The thermal expansion coefficient along this direction is larger than that perpendicular to it as shown in the inset of Supplementary Figure 9, thus under cooling, the sample is subjected to a symmetry-breaking

Raman response and Lehmann representation
We start from the simple model layed out in the main text, where two degenerate harmonic modes (phonons) with E g symmetry couple to an Ising variable |s〉∈{| ⇑〉=(1, 0), | ⇓〉=(0, 1)} capturing the nematic degree of freedom. In addition, the systems' degeneracy can be externally lifted by coupling to external stress σ ext that plays the role of a a conjugate field to the Ising variable, via −σ ext τ z Here and below, the Pauli matrices τ α act in the space of the pseudospin |s〉. Using the approach of Ref. 7, the Raman response in the B 1g is given by For the E g modes we want to determine which are the Fourier transform of with α = x, y. Without nematic order the two E g Green's functions are identical.
In the coupled problem it is better to use the Lehmann representation of the Green's function and determine the eigenvalues and matrix elements. To this end we consider a generic Green's ]〉, and introduce the Fourier transform of the correlation functions After a bit of algebra follows where {|ℓ〉} are the exact eigenfunctions of the Hamiltonian with eigenvalues {E ℓ }, i.e. H|ℓ〉 = E ℓ |ℓ〉 and Z = ℓ e −β E ℓ is the partition sum. In our case holds A = B and get access to the propagator after determining the spectral function since J(ω) = −2n(ω)ImG r AA (ω).